Notes on Metrics, Measures, and Dimensions

Preface In this monograph various notions of measure and content for subsets of metric spaces are considered, and both metric and topological notions of dimension. Some familiarity with basic topics in analysis such as Riemann integrals, metric spaces, and continuous functions is assumed, as in [50, 88, 160], for instance, although a number of definitions and results that are needed are reviewed in the text. More sophisticated matters such as Lebesgue integrals are not required, but can be complementary. It is hoped that the present endeavor might be useful in a supplementary role for a course, in conjunction with other material at the discretion of the instructor. It is intended that the presentation be relatively brief and informal, sometimes leaving as an excercise to the reader the task of filling in some details. ii Contents Preface ii 1 Metric spaces 1 1.